Properties

Label 112.4.i.c.81.1
Level $112$
Weight $4$
Character 112.81
Analytic conductor $6.608$
Analytic rank $0$
Dimension $2$
Inner twists $2$

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Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms
 
[N,k,chi] = [112,4,Mod(65,112)]
 
mf = mfinit([N,k,chi],0)
 
lf = mfeigenbasis(mf)
 
from sage.modular.dirichlet import DirichletCharacter
 
H = DirichletGroup(112, base_ring=CyclotomicField(6))
 
chi = DirichletCharacter(H, H._module([0, 0, 2]))
 
N = Newforms(chi, 4, names="a")
 
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
 
chi := DirichletCharacter("112.65");
 
S:= CuspForms(chi, 4);
 
N := Newforms(S);
 
Level: \( N \) \(=\) \( 112 = 2^{4} \cdot 7 \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 112.i (of order \(3\), degree \(2\), not minimal)

Newform invariants

comment: select newform
 
sage: f = N[0] # Warning: the index may be different
 
gp: f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: \(6.60821392064\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{6})\)
comment: defining polynomial
 
gp: f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{2} - x + 1 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{9}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 7)
Sato-Tate group: $\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label 81.1
Root \(0.500000 + 0.866025i\) of defining polynomial
Character \(\chi\) \(=\) 112.81
Dual form 112.4.i.c.65.1

$q$-expansion

comment: q-expansion
 
sage: f.q_expansion() # note that sage often uses an isomorphic number field
 
gp: mfcoefs(f, 20)
 
\(f(q)\) \(=\) \(q+(3.50000 - 6.06218i) q^{3} +(-3.50000 - 6.06218i) q^{5} +(-14.0000 - 12.1244i) q^{7} +(-11.0000 - 19.0526i) q^{9} +(-2.50000 + 4.33013i) q^{11} -14.0000 q^{13} -49.0000 q^{15} +(10.5000 - 18.1865i) q^{17} +(24.5000 + 42.4352i) q^{19} +(-122.500 + 42.4352i) q^{21} +(-79.5000 - 137.698i) q^{23} +(38.0000 - 65.8179i) q^{25} +35.0000 q^{27} +58.0000 q^{29} +(73.5000 - 127.306i) q^{31} +(17.5000 + 30.3109i) q^{33} +(-24.5000 + 127.306i) q^{35} +(-109.500 - 189.660i) q^{37} +(-49.0000 + 84.8705i) q^{39} +350.000 q^{41} +124.000 q^{43} +(-77.0000 + 133.368i) q^{45} +(262.500 + 454.663i) q^{47} +(49.0000 + 339.482i) q^{49} +(-73.5000 - 127.306i) q^{51} +(-151.500 + 262.406i) q^{53} +35.0000 q^{55} +343.000 q^{57} +(-52.5000 + 90.9327i) q^{59} +(206.500 + 357.668i) q^{61} +(-77.0000 + 400.104i) q^{63} +(49.0000 + 84.8705i) q^{65} +(207.500 - 359.401i) q^{67} -1113.00 q^{69} +432.000 q^{71} +(556.500 - 963.886i) q^{73} +(-266.000 - 460.726i) q^{75} +(87.5000 - 30.3109i) q^{77} +(-51.5000 - 89.2006i) q^{79} +(419.500 - 726.595i) q^{81} -1092.00 q^{83} -147.000 q^{85} +(203.000 - 351.606i) q^{87} +(164.500 + 284.922i) q^{89} +(196.000 + 169.741i) q^{91} +(-514.500 - 891.140i) q^{93} +(171.500 - 297.047i) q^{95} -882.000 q^{97} +110.000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 7 q^{3} - 7 q^{5} - 28 q^{7} - 22 q^{9} - 5 q^{11} - 28 q^{13} - 98 q^{15} + 21 q^{17} + 49 q^{19} - 245 q^{21} - 159 q^{23} + 76 q^{25} + 70 q^{27} + 116 q^{29} + 147 q^{31} + 35 q^{33} - 49 q^{35}+ \cdots + 220 q^{99}+O(q^{100}) \) Copy content Toggle raw display

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/112\mathbb{Z}\right)^\times\).

\(n\) \(15\) \(17\) \(85\)
\(\chi(n)\) \(1\) \(e\left(\frac{2}{3}\right)\) \(1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).



Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 0 0
\(3\) 3.50000 6.06218i 0.673575 1.16667i −0.303308 0.952893i \(-0.598091\pi\)
0.976883 0.213774i \(-0.0685756\pi\)
\(4\) 0 0
\(5\) −3.50000 6.06218i −0.313050 0.542218i 0.665971 0.745977i \(-0.268017\pi\)
−0.979021 + 0.203760i \(0.934684\pi\)
\(6\) 0 0
\(7\) −14.0000 12.1244i −0.755929 0.654654i
\(8\) 0 0
\(9\) −11.0000 19.0526i −0.407407 0.705650i
\(10\) 0 0
\(11\) −2.50000 + 4.33013i −0.0685253 + 0.118689i −0.898252 0.439480i \(-0.855163\pi\)
0.829727 + 0.558169i \(0.188496\pi\)
\(12\) 0 0
\(13\) −14.0000 −0.298685 −0.149342 0.988786i \(-0.547716\pi\)
−0.149342 + 0.988786i \(0.547716\pi\)
\(14\) 0 0
\(15\) −49.0000 −0.843450
\(16\) 0 0
\(17\) 10.5000 18.1865i 0.149801 0.259464i −0.781353 0.624090i \(-0.785470\pi\)
0.931154 + 0.364626i \(0.118803\pi\)
\(18\) 0 0
\(19\) 24.5000 + 42.4352i 0.295826 + 0.512385i 0.975177 0.221429i \(-0.0710720\pi\)
−0.679351 + 0.733813i \(0.737739\pi\)
\(20\) 0 0
\(21\) −122.500 + 42.4352i −1.27294 + 0.440959i
\(22\) 0 0
\(23\) −79.5000 137.698i −0.720735 1.24835i −0.960706 0.277569i \(-0.910471\pi\)
0.239971 0.970780i \(-0.422862\pi\)
\(24\) 0 0
\(25\) 38.0000 65.8179i 0.304000 0.526543i
\(26\) 0 0
\(27\) 35.0000 0.249472
\(28\) 0 0
\(29\) 58.0000 0.371391 0.185695 0.982607i \(-0.440546\pi\)
0.185695 + 0.982607i \(0.440546\pi\)
\(30\) 0 0
\(31\) 73.5000 127.306i 0.425838 0.737574i −0.570660 0.821186i \(-0.693313\pi\)
0.996498 + 0.0836128i \(0.0266459\pi\)
\(32\) 0 0
\(33\) 17.5000 + 30.3109i 0.0923139 + 0.159892i
\(34\) 0 0
\(35\) −24.5000 + 127.306i −0.118322 + 0.614817i
\(36\) 0 0
\(37\) −109.500 189.660i −0.486532 0.842698i 0.513348 0.858181i \(-0.328405\pi\)
−0.999880 + 0.0154821i \(0.995072\pi\)
\(38\) 0 0
\(39\) −49.0000 + 84.8705i −0.201187 + 0.348466i
\(40\) 0 0
\(41\) 350.000 1.33319 0.666595 0.745420i \(-0.267751\pi\)
0.666595 + 0.745420i \(0.267751\pi\)
\(42\) 0 0
\(43\) 124.000 0.439763 0.219882 0.975527i \(-0.429433\pi\)
0.219882 + 0.975527i \(0.429433\pi\)
\(44\) 0 0
\(45\) −77.0000 + 133.368i −0.255077 + 0.441807i
\(46\) 0 0
\(47\) 262.500 + 454.663i 0.814671 + 1.41105i 0.909564 + 0.415565i \(0.136416\pi\)
−0.0948921 + 0.995488i \(0.530251\pi\)
\(48\) 0 0
\(49\) 49.0000 + 339.482i 0.142857 + 0.989743i
\(50\) 0 0
\(51\) −73.5000 127.306i −0.201805 0.349537i
\(52\) 0 0
\(53\) −151.500 + 262.406i −0.392644 + 0.680079i −0.992797 0.119806i \(-0.961773\pi\)
0.600153 + 0.799885i \(0.295106\pi\)
\(54\) 0 0
\(55\) 35.0000 0.0858073
\(56\) 0 0
\(57\) 343.000 0.797043
\(58\) 0 0
\(59\) −52.5000 + 90.9327i −0.115846 + 0.200651i −0.918118 0.396308i \(-0.870291\pi\)
0.802272 + 0.596959i \(0.203625\pi\)
\(60\) 0 0
\(61\) 206.500 + 357.668i 0.433436 + 0.750734i 0.997167 0.0752252i \(-0.0239676\pi\)
−0.563730 + 0.825959i \(0.690634\pi\)
\(62\) 0 0
\(63\) −77.0000 + 400.104i −0.153986 + 0.800132i
\(64\) 0 0
\(65\) 49.0000 + 84.8705i 0.0935031 + 0.161952i
\(66\) 0 0
\(67\) 207.500 359.401i 0.378361 0.655340i −0.612463 0.790499i \(-0.709821\pi\)
0.990824 + 0.135159i \(0.0431546\pi\)
\(68\) 0 0
\(69\) −1113.00 −1.94188
\(70\) 0 0
\(71\) 432.000 0.722098 0.361049 0.932547i \(-0.382419\pi\)
0.361049 + 0.932547i \(0.382419\pi\)
\(72\) 0 0
\(73\) 556.500 963.886i 0.892238 1.54540i 0.0550526 0.998483i \(-0.482467\pi\)
0.837186 0.546919i \(-0.184199\pi\)
\(74\) 0 0
\(75\) −266.000 460.726i −0.409534 0.709333i
\(76\) 0 0
\(77\) 87.5000 30.3109i 0.129501 0.0448603i
\(78\) 0 0
\(79\) −51.5000 89.2006i −0.0733443 0.127036i 0.827021 0.562171i \(-0.190034\pi\)
−0.900365 + 0.435135i \(0.856701\pi\)
\(80\) 0 0
\(81\) 419.500 726.595i 0.575446 0.996701i
\(82\) 0 0
\(83\) −1092.00 −1.44413 −0.722064 0.691827i \(-0.756806\pi\)
−0.722064 + 0.691827i \(0.756806\pi\)
\(84\) 0 0
\(85\) −147.000 −0.187581
\(86\) 0 0
\(87\) 203.000 351.606i 0.250160 0.433289i
\(88\) 0 0
\(89\) 164.500 + 284.922i 0.195921 + 0.339345i 0.947202 0.320637i \(-0.103897\pi\)
−0.751281 + 0.659982i \(0.770564\pi\)
\(90\) 0 0
\(91\) 196.000 + 169.741i 0.225784 + 0.195535i
\(92\) 0 0
\(93\) −514.500 891.140i −0.573668 0.993623i
\(94\) 0 0
\(95\) 171.500 297.047i 0.185216 0.320804i
\(96\) 0 0
\(97\) −882.000 −0.923232 −0.461616 0.887080i \(-0.652730\pi\)
−0.461616 + 0.887080i \(0.652730\pi\)
\(98\) 0 0
\(99\) 110.000 0.111671
\(100\) 0 0
\(101\) −689.500 + 1194.25i −0.679285 + 1.17656i 0.295911 + 0.955215i \(0.404377\pi\)
−0.975196 + 0.221341i \(0.928957\pi\)
\(102\) 0 0
\(103\) −339.500 588.031i −0.324776 0.562529i 0.656691 0.754160i \(-0.271956\pi\)
−0.981467 + 0.191631i \(0.938622\pi\)
\(104\) 0 0
\(105\) 686.000 + 594.093i 0.637588 + 0.552167i
\(106\) 0 0
\(107\) 228.500 + 395.774i 0.206448 + 0.357578i 0.950593 0.310440i \(-0.100476\pi\)
−0.744145 + 0.668018i \(0.767143\pi\)
\(108\) 0 0
\(109\) 562.500 974.279i 0.494291 0.856137i −0.505687 0.862717i \(-0.668761\pi\)
0.999978 + 0.00657959i \(0.00209436\pi\)
\(110\) 0 0
\(111\) −1533.00 −1.31086
\(112\) 0 0
\(113\) −1538.00 −1.28038 −0.640190 0.768217i \(-0.721144\pi\)
−0.640190 + 0.768217i \(0.721144\pi\)
\(114\) 0 0
\(115\) −556.500 + 963.886i −0.451251 + 0.781590i
\(116\) 0 0
\(117\) 154.000 + 266.736i 0.121686 + 0.210767i
\(118\) 0 0
\(119\) −367.500 + 127.306i −0.283098 + 0.0980680i
\(120\) 0 0
\(121\) 653.000 + 1131.03i 0.490609 + 0.849759i
\(122\) 0 0
\(123\) 1225.00 2121.76i 0.898004 1.55539i
\(124\) 0 0
\(125\) −1407.00 −1.00677
\(126\) 0 0
\(127\) −72.0000 −0.0503068 −0.0251534 0.999684i \(-0.508007\pi\)
−0.0251534 + 0.999684i \(0.508007\pi\)
\(128\) 0 0
\(129\) 434.000 751.710i 0.296214 0.513057i
\(130\) 0 0
\(131\) 1074.50 + 1861.09i 0.716637 + 1.24125i 0.962325 + 0.271903i \(0.0876531\pi\)
−0.245687 + 0.969349i \(0.579014\pi\)
\(132\) 0 0
\(133\) 171.500 891.140i 0.111812 0.580990i
\(134\) 0 0
\(135\) −122.500 212.176i −0.0780972 0.135268i
\(136\) 0 0
\(137\) 562.500 974.279i 0.350786 0.607578i −0.635602 0.772017i \(-0.719248\pi\)
0.986387 + 0.164439i \(0.0525813\pi\)
\(138\) 0 0
\(139\) −252.000 −0.153772 −0.0768862 0.997040i \(-0.524498\pi\)
−0.0768862 + 0.997040i \(0.524498\pi\)
\(140\) 0 0
\(141\) 3675.00 2.19497
\(142\) 0 0
\(143\) 35.0000 60.6218i 0.0204675 0.0354507i
\(144\) 0 0
\(145\) −203.000 351.606i −0.116264 0.201375i
\(146\) 0 0
\(147\) 2229.50 + 891.140i 1.25093 + 0.500000i
\(148\) 0 0
\(149\) 100.500 + 174.071i 0.0552569 + 0.0957078i 0.892331 0.451382i \(-0.149069\pi\)
−0.837074 + 0.547090i \(0.815736\pi\)
\(150\) 0 0
\(151\) 809.500 1402.10i 0.436266 0.755635i −0.561132 0.827726i \(-0.689634\pi\)
0.997398 + 0.0720914i \(0.0229673\pi\)
\(152\) 0 0
\(153\) −462.000 −0.244121
\(154\) 0 0
\(155\) −1029.00 −0.533234
\(156\) 0 0
\(157\) −339.500 + 588.031i −0.172580 + 0.298917i −0.939321 0.343039i \(-0.888544\pi\)
0.766741 + 0.641956i \(0.221877\pi\)
\(158\) 0 0
\(159\) 1060.50 + 1836.84i 0.528950 + 0.916169i
\(160\) 0 0
\(161\) −556.500 + 2891.66i −0.272412 + 1.41549i
\(162\) 0 0
\(163\) −233.500 404.434i −0.112203 0.194342i 0.804455 0.594014i \(-0.202457\pi\)
−0.916658 + 0.399672i \(0.869124\pi\)
\(164\) 0 0
\(165\) 122.500 212.176i 0.0577976 0.100108i
\(166\) 0 0
\(167\) −1204.00 −0.557894 −0.278947 0.960306i \(-0.589985\pi\)
−0.278947 + 0.960306i \(0.589985\pi\)
\(168\) 0 0
\(169\) −2001.00 −0.910787
\(170\) 0 0
\(171\) 539.000 933.575i 0.241043 0.417499i
\(172\) 0 0
\(173\) 1410.50 + 2443.06i 0.619875 + 1.07365i 0.989508 + 0.144477i \(0.0461499\pi\)
−0.369633 + 0.929178i \(0.620517\pi\)
\(174\) 0 0
\(175\) −1330.00 + 460.726i −0.574506 + 0.199015i
\(176\) 0 0
\(177\) 367.500 + 636.529i 0.156062 + 0.270307i
\(178\) 0 0
\(179\) −1626.50 + 2817.18i −0.679164 + 1.17635i 0.296069 + 0.955166i \(0.404324\pi\)
−0.975233 + 0.221180i \(0.929009\pi\)
\(180\) 0 0
\(181\) 1582.00 0.649664 0.324832 0.945772i \(-0.394692\pi\)
0.324832 + 0.945772i \(0.394692\pi\)
\(182\) 0 0
\(183\) 2891.00 1.16781
\(184\) 0 0
\(185\) −766.500 + 1327.62i −0.304617 + 0.527613i
\(186\) 0 0
\(187\) 52.5000 + 90.9327i 0.0205304 + 0.0355597i
\(188\) 0 0
\(189\) −490.000 424.352i −0.188583 0.163318i
\(190\) 0 0
\(191\) 1278.50 + 2214.43i 0.484340 + 0.838902i 0.999838 0.0179887i \(-0.00572630\pi\)
−0.515498 + 0.856891i \(0.672393\pi\)
\(192\) 0 0
\(193\) 198.500 343.812i 0.0740329 0.128229i −0.826632 0.562742i \(-0.809746\pi\)
0.900665 + 0.434514i \(0.143080\pi\)
\(194\) 0 0
\(195\) 686.000 0.251926
\(196\) 0 0
\(197\) 2914.00 1.05388 0.526939 0.849903i \(-0.323340\pi\)
0.526939 + 0.849903i \(0.323340\pi\)
\(198\) 0 0
\(199\) 1669.50 2891.66i 0.594712 1.03007i −0.398875 0.917005i \(-0.630599\pi\)
0.993587 0.113066i \(-0.0360673\pi\)
\(200\) 0 0
\(201\) −1452.50 2515.80i −0.509709 0.882841i
\(202\) 0 0
\(203\) −812.000 703.213i −0.280745 0.243132i
\(204\) 0 0
\(205\) −1225.00 2121.76i −0.417355 0.722880i
\(206\) 0 0
\(207\) −1749.00 + 3029.36i −0.587265 + 1.01717i
\(208\) 0 0
\(209\) −245.000 −0.0810861
\(210\) 0 0
\(211\) −1780.00 −0.580759 −0.290380 0.956911i \(-0.593782\pi\)
−0.290380 + 0.956911i \(0.593782\pi\)
\(212\) 0 0
\(213\) 1512.00 2618.86i 0.486387 0.842448i
\(214\) 0 0
\(215\) −434.000 751.710i −0.137668 0.238447i
\(216\) 0 0
\(217\) −2572.50 + 891.140i −0.804759 + 0.278777i
\(218\) 0 0
\(219\) −3895.50 6747.20i −1.20198 2.08189i
\(220\) 0 0
\(221\) −147.000 + 254.611i −0.0447434 + 0.0774978i
\(222\) 0 0
\(223\) 1400.00 0.420408 0.210204 0.977658i \(-0.432587\pi\)
0.210204 + 0.977658i \(0.432587\pi\)
\(224\) 0 0
\(225\) −1672.00 −0.495407
\(226\) 0 0
\(227\) −1102.50 + 1909.59i −0.322359 + 0.558342i −0.980974 0.194138i \(-0.937809\pi\)
0.658615 + 0.752480i \(0.271142\pi\)
\(228\) 0 0
\(229\) −143.500 248.549i −0.0414094 0.0717231i 0.844578 0.535433i \(-0.179851\pi\)
−0.885987 + 0.463710i \(0.846518\pi\)
\(230\) 0 0
\(231\) 122.500 636.529i 0.0348914 0.181301i
\(232\) 0 0
\(233\) −2293.50 3972.46i −0.644859 1.11693i −0.984334 0.176314i \(-0.943583\pi\)
0.339475 0.940615i \(-0.389751\pi\)
\(234\) 0 0
\(235\) 1837.50 3182.64i 0.510065 0.883459i
\(236\) 0 0
\(237\) −721.000 −0.197612
\(238\) 0 0
\(239\) −1668.00 −0.451439 −0.225720 0.974192i \(-0.572473\pi\)
−0.225720 + 0.974192i \(0.572473\pi\)
\(240\) 0 0
\(241\) 1704.50 2952.28i 0.455587 0.789100i −0.543135 0.839646i \(-0.682763\pi\)
0.998722 + 0.0505456i \(0.0160960\pi\)
\(242\) 0 0
\(243\) −2464.00 4267.77i −0.650476 1.12666i
\(244\) 0 0
\(245\) 1886.50 1485.23i 0.491935 0.387298i
\(246\) 0 0
\(247\) −343.000 594.093i −0.0883586 0.153042i
\(248\) 0 0
\(249\) −3822.00 + 6619.90i −0.972729 + 1.68482i
\(250\) 0 0
\(251\) 4760.00 1.19701 0.598503 0.801121i \(-0.295762\pi\)
0.598503 + 0.801121i \(0.295762\pi\)
\(252\) 0 0
\(253\) 795.000 0.197554
\(254\) 0 0
\(255\) −514.500 + 891.140i −0.126350 + 0.218845i
\(256\) 0 0
\(257\) 402.500 + 697.150i 0.0976936 + 0.169210i 0.910730 0.413003i \(-0.135520\pi\)
−0.813036 + 0.582213i \(0.802187\pi\)
\(258\) 0 0
\(259\) −766.500 + 3982.85i −0.183892 + 0.955530i
\(260\) 0 0
\(261\) −638.000 1105.05i −0.151307 0.262072i
\(262\) 0 0
\(263\) −128.500 + 222.569i −0.0301279 + 0.0521831i −0.880696 0.473681i \(-0.842925\pi\)
0.850568 + 0.525865i \(0.176258\pi\)
\(264\) 0 0
\(265\) 2121.00 0.491668
\(266\) 0 0
\(267\) 2303.00 0.527870
\(268\) 0 0
\(269\) −1795.50 + 3109.90i −0.406965 + 0.704884i −0.994548 0.104280i \(-0.966746\pi\)
0.587583 + 0.809164i \(0.300080\pi\)
\(270\) 0 0
\(271\) 696.500 + 1206.37i 0.156123 + 0.270413i 0.933467 0.358662i \(-0.116767\pi\)
−0.777344 + 0.629075i \(0.783434\pi\)
\(272\) 0 0
\(273\) 1715.00 594.093i 0.380207 0.131708i
\(274\) 0 0
\(275\) 190.000 + 329.090i 0.0416634 + 0.0721631i
\(276\) 0 0
\(277\) −207.500 + 359.401i −0.0450089 + 0.0779577i −0.887652 0.460514i \(-0.847665\pi\)
0.842643 + 0.538472i \(0.180998\pi\)
\(278\) 0 0
\(279\) −3234.00 −0.693959
\(280\) 0 0
\(281\) −4954.00 −1.05171 −0.525856 0.850574i \(-0.676255\pi\)
−0.525856 + 0.850574i \(0.676255\pi\)
\(282\) 0 0
\(283\) −2138.50 + 3703.99i −0.449190 + 0.778019i −0.998333 0.0577087i \(-0.981621\pi\)
0.549144 + 0.835728i \(0.314954\pi\)
\(284\) 0 0
\(285\) −1200.50 2079.33i −0.249514 0.432171i
\(286\) 0 0
\(287\) −4900.00 4243.52i −1.00780 0.872778i
\(288\) 0 0
\(289\) 2236.00 + 3872.87i 0.455119 + 0.788289i
\(290\) 0 0
\(291\) −3087.00 + 5346.84i −0.621866 + 1.07710i
\(292\) 0 0
\(293\) 7742.00 1.54366 0.771830 0.635829i \(-0.219342\pi\)
0.771830 + 0.635829i \(0.219342\pi\)
\(294\) 0 0
\(295\) 735.000 0.145062
\(296\) 0 0
\(297\) −87.5000 + 151.554i −0.0170952 + 0.0296097i
\(298\) 0 0
\(299\) 1113.00 + 1927.77i 0.215272 + 0.372863i
\(300\) 0 0
\(301\) −1736.00 1503.42i −0.332430 0.287893i
\(302\) 0 0
\(303\) 4826.50 + 8359.74i 0.915100 + 1.58500i
\(304\) 0 0
\(305\) 1445.50 2503.68i 0.271374 0.470034i
\(306\) 0 0
\(307\) 7364.00 1.36901 0.684504 0.729009i \(-0.260019\pi\)
0.684504 + 0.729009i \(0.260019\pi\)
\(308\) 0 0
\(309\) −4753.00 −0.875044
\(310\) 0 0
\(311\) 4987.50 8638.60i 0.909374 1.57508i 0.0944372 0.995531i \(-0.469895\pi\)
0.814936 0.579550i \(-0.196772\pi\)
\(312\) 0 0
\(313\) 2376.50 + 4116.22i 0.429162 + 0.743330i 0.996799 0.0799485i \(-0.0254756\pi\)
−0.567637 + 0.823279i \(0.692142\pi\)
\(314\) 0 0
\(315\) 2695.00 933.575i 0.482051 0.166987i
\(316\) 0 0
\(317\) 1738.50 + 3011.17i 0.308025 + 0.533515i 0.977930 0.208932i \(-0.0669987\pi\)
−0.669905 + 0.742447i \(0.733665\pi\)
\(318\) 0 0
\(319\) −145.000 + 251.147i −0.0254497 + 0.0440801i
\(320\) 0 0
\(321\) 3199.00 0.556233
\(322\) 0 0
\(323\) 1029.00 0.177260
\(324\) 0 0
\(325\) −532.000 + 921.451i −0.0908002 + 0.157270i
\(326\) 0 0
\(327\) −3937.50 6819.95i −0.665885 1.15335i
\(328\) 0 0
\(329\) 1837.50 9547.93i 0.307917 1.59998i
\(330\) 0 0
\(331\) 1670.50 + 2893.39i 0.277399 + 0.480469i 0.970738 0.240143i \(-0.0771944\pi\)
−0.693339 + 0.720612i \(0.743861\pi\)
\(332\) 0 0
\(333\) −2409.00 + 4172.51i −0.396434 + 0.686643i
\(334\) 0 0
\(335\) −2905.00 −0.473782
\(336\) 0 0
\(337\) 7366.00 1.19066 0.595329 0.803482i \(-0.297022\pi\)
0.595329 + 0.803482i \(0.297022\pi\)
\(338\) 0 0
\(339\) −5383.00 + 9323.63i −0.862432 + 1.49378i
\(340\) 0 0
\(341\) 367.500 + 636.529i 0.0583614 + 0.101085i
\(342\) 0 0
\(343\) 3430.00 5346.84i 0.539949 0.841698i
\(344\) 0 0
\(345\) 3895.50 + 6747.20i 0.607903 + 1.05292i
\(346\) 0 0
\(347\) 3707.50 6421.58i 0.573571 0.993454i −0.422625 0.906305i \(-0.638891\pi\)
0.996195 0.0871487i \(-0.0277755\pi\)
\(348\) 0 0
\(349\) −3878.00 −0.594798 −0.297399 0.954753i \(-0.596119\pi\)
−0.297399 + 0.954753i \(0.596119\pi\)
\(350\) 0 0
\(351\) −490.000 −0.0745136
\(352\) 0 0
\(353\) −633.500 + 1097.25i −0.0955179 + 0.165442i −0.909825 0.414993i \(-0.863784\pi\)
0.814307 + 0.580435i \(0.197117\pi\)
\(354\) 0 0
\(355\) −1512.00 2618.86i −0.226052 0.391534i
\(356\) 0 0
\(357\) −514.500 + 2673.42i −0.0762751 + 0.396337i
\(358\) 0 0
\(359\) 2342.50 + 4057.33i 0.344380 + 0.596484i 0.985241 0.171173i \(-0.0547558\pi\)
−0.640861 + 0.767657i \(0.721422\pi\)
\(360\) 0 0
\(361\) 2229.00 3860.74i 0.324974 0.562872i
\(362\) 0 0
\(363\) 9142.00 1.32185
\(364\) 0 0
\(365\) −7791.00 −1.11726
\(366\) 0 0
\(367\) −2320.50 + 4019.22i −0.330052 + 0.571667i −0.982522 0.186148i \(-0.940400\pi\)
0.652470 + 0.757815i \(0.273733\pi\)
\(368\) 0 0
\(369\) −3850.00 6668.40i −0.543152 0.940766i
\(370\) 0 0
\(371\) 5302.50 1836.84i 0.742027 0.257046i
\(372\) 0 0
\(373\) 4398.50 + 7618.43i 0.610578 + 1.05755i 0.991143 + 0.132798i \(0.0423963\pi\)
−0.380565 + 0.924754i \(0.624270\pi\)
\(374\) 0 0
\(375\) −4924.50 + 8529.48i −0.678134 + 1.17456i
\(376\) 0 0
\(377\) −812.000 −0.110929
\(378\) 0 0
\(379\) −13680.0 −1.85407 −0.927037 0.374969i \(-0.877653\pi\)
−0.927037 + 0.374969i \(0.877653\pi\)
\(380\) 0 0
\(381\) −252.000 + 436.477i −0.0338854 + 0.0586913i
\(382\) 0 0
\(383\) 4882.50 + 8456.74i 0.651395 + 1.12825i 0.982785 + 0.184755i \(0.0591490\pi\)
−0.331390 + 0.943494i \(0.607518\pi\)
\(384\) 0 0
\(385\) −490.000 424.352i −0.0648642 0.0561740i
\(386\) 0 0
\(387\) −1364.00 2362.52i −0.179163 0.310319i
\(388\) 0 0
\(389\) −865.500 + 1499.09i −0.112809 + 0.195390i −0.916902 0.399113i \(-0.869318\pi\)
0.804093 + 0.594504i \(0.202651\pi\)
\(390\) 0 0
\(391\) −3339.00 −0.431868
\(392\) 0 0
\(393\) 15043.0 1.93084
\(394\) 0 0
\(395\) −360.500 + 624.404i −0.0459208 + 0.0795372i
\(396\) 0 0
\(397\) −5491.50 9511.56i −0.694233 1.20245i −0.970439 0.241348i \(-0.922410\pi\)
0.276206 0.961099i \(-0.410923\pi\)
\(398\) 0 0
\(399\) −4802.00 4158.65i −0.602508 0.521787i
\(400\) 0 0
\(401\) −3301.50 5718.37i −0.411145 0.712124i 0.583870 0.811847i \(-0.301538\pi\)
−0.995015 + 0.0997232i \(0.968204\pi\)
\(402\) 0 0
\(403\) −1029.00 + 1782.28i −0.127191 + 0.220302i
\(404\) 0 0
\(405\) −5873.00 −0.720572
\(406\) 0 0
\(407\) 1095.00 0.133359
\(408\) 0 0
\(409\) −5477.50 + 9487.31i −0.662213 + 1.14699i 0.317820 + 0.948151i \(0.397049\pi\)
−0.980033 + 0.198835i \(0.936284\pi\)
\(410\) 0 0
\(411\) −3937.50 6819.95i −0.472561 0.818500i
\(412\) 0 0
\(413\) 1837.50 636.529i 0.218928 0.0758391i
\(414\) 0 0
\(415\) 3822.00 + 6619.90i 0.452083 + 0.783031i
\(416\) 0 0
\(417\) −882.000 + 1527.67i −0.103577 + 0.179401i
\(418\) 0 0
\(419\) −6636.00 −0.773723 −0.386861 0.922138i \(-0.626441\pi\)
−0.386861 + 0.922138i \(0.626441\pi\)
\(420\) 0 0
\(421\) −16630.0 −1.92517 −0.962585 0.270980i \(-0.912652\pi\)
−0.962585 + 0.270980i \(0.912652\pi\)
\(422\) 0 0
\(423\) 5775.00 10002.6i 0.663806 1.14975i
\(424\) 0 0
\(425\) −798.000 1382.18i −0.0910793 0.157754i
\(426\) 0 0
\(427\) 1445.50 7511.04i 0.163824 0.851252i
\(428\) 0 0
\(429\) −245.000 424.352i −0.0275728 0.0477574i
\(430\) 0 0
\(431\) 2461.50 4263.44i 0.275096 0.476480i −0.695064 0.718948i \(-0.744624\pi\)
0.970159 + 0.242468i \(0.0779571\pi\)
\(432\) 0 0
\(433\) 8974.00 0.995988 0.497994 0.867180i \(-0.334070\pi\)
0.497994 + 0.867180i \(0.334070\pi\)
\(434\) 0 0
\(435\) −2842.00 −0.313249
\(436\) 0 0
\(437\) 3895.50 6747.20i 0.426423 0.738587i
\(438\) 0 0
\(439\) −2089.50 3619.12i −0.227167 0.393465i 0.729800 0.683660i \(-0.239613\pi\)
−0.956967 + 0.290195i \(0.906280\pi\)
\(440\) 0 0
\(441\) 5929.00 4667.88i 0.640212 0.504036i
\(442\) 0 0
\(443\) −6463.50 11195.1i −0.693206 1.20067i −0.970782 0.239964i \(-0.922864\pi\)
0.277576 0.960704i \(-0.410469\pi\)
\(444\) 0 0
\(445\) 1151.50 1994.46i 0.122666 0.212464i
\(446\) 0 0
\(447\) 1407.00 0.148879
\(448\) 0 0
\(449\) −2826.00 −0.297032 −0.148516 0.988910i \(-0.547450\pi\)
−0.148516 + 0.988910i \(0.547450\pi\)
\(450\) 0 0
\(451\) −875.000 + 1515.54i −0.0913573 + 0.158235i
\(452\) 0 0
\(453\) −5666.50 9814.67i −0.587716 1.01795i
\(454\) 0 0
\(455\) 343.000 1782.28i 0.0353409 0.183636i
\(456\) 0 0
\(457\) −4239.50 7343.03i −0.433951 0.751625i 0.563259 0.826281i \(-0.309547\pi\)
−0.997209 + 0.0746560i \(0.976214\pi\)
\(458\) 0 0
\(459\) 367.500 636.529i 0.0373713 0.0647290i
\(460\) 0 0
\(461\) 9338.00 0.943414 0.471707 0.881755i \(-0.343638\pi\)
0.471707 + 0.881755i \(0.343638\pi\)
\(462\) 0 0
\(463\) 4016.00 0.403109 0.201554 0.979477i \(-0.435401\pi\)
0.201554 + 0.979477i \(0.435401\pi\)
\(464\) 0 0
\(465\) −3601.50 + 6237.98i −0.359173 + 0.622106i
\(466\) 0 0
\(467\) −2929.50 5074.04i −0.290281 0.502781i 0.683595 0.729861i \(-0.260415\pi\)
−0.973876 + 0.227080i \(0.927082\pi\)
\(468\) 0 0
\(469\) −7262.50 + 2515.80i −0.715034 + 0.247695i
\(470\) 0 0
\(471\) 2376.50 + 4116.22i 0.232491 + 0.402687i
\(472\) 0 0
\(473\) −310.000 + 536.936i −0.0301349 + 0.0521952i
\(474\) 0 0
\(475\) 3724.00 0.359724
\(476\) 0 0
\(477\) 6666.00 0.639864
\(478\) 0 0
\(479\) 3251.50 5631.76i 0.310156 0.537206i −0.668240 0.743946i \(-0.732952\pi\)
0.978396 + 0.206740i \(0.0662853\pi\)
\(480\) 0 0
\(481\) 1533.00 + 2655.23i 0.145320 + 0.251701i
\(482\) 0 0
\(483\) 15582.0 + 13494.4i 1.46792 + 1.27126i
\(484\) 0 0
\(485\) 3087.00 + 5346.84i 0.289017 + 0.500593i
\(486\) 0 0
\(487\) −8024.50 + 13898.8i −0.746663 + 1.29326i 0.202751 + 0.979230i \(0.435012\pi\)
−0.949414 + 0.314028i \(0.898322\pi\)
\(488\) 0 0
\(489\) −3269.00 −0.302309
\(490\) 0 0
\(491\) −8864.00 −0.814718 −0.407359 0.913268i \(-0.633550\pi\)
−0.407359 + 0.913268i \(0.633550\pi\)
\(492\) 0 0
\(493\) 609.000 1054.82i 0.0556348 0.0963624i
\(494\) 0 0
\(495\) −385.000 666.840i −0.0349585 0.0605499i
\(496\) 0 0
\(497\) −6048.00 5237.72i −0.545855 0.472724i
\(498\) 0 0
\(499\) −5105.50 8842.99i −0.458023 0.793319i 0.540833 0.841130i \(-0.318109\pi\)
−0.998856 + 0.0478104i \(0.984776\pi\)
\(500\) 0 0
\(501\) −4214.00 + 7298.86i −0.375784 + 0.650876i
\(502\) 0 0
\(503\) 1680.00 0.148921 0.0744607 0.997224i \(-0.476276\pi\)
0.0744607 + 0.997224i \(0.476276\pi\)
\(504\) 0 0
\(505\) 9653.00 0.850600
\(506\) 0 0
\(507\) −7003.50 + 12130.4i −0.613484 + 1.06259i
\(508\) 0 0
\(509\) 4728.50 + 8190.00i 0.411762 + 0.713193i 0.995083 0.0990489i \(-0.0315800\pi\)
−0.583320 + 0.812242i \(0.698247\pi\)
\(510\) 0 0
\(511\) −19477.5 + 6747.20i −1.68617 + 0.584107i
\(512\) 0 0
\(513\) 857.500 + 1485.23i 0.0738003 + 0.127826i
\(514\) 0 0
\(515\) −2376.50 + 4116.22i −0.203342 + 0.352199i
\(516\) 0 0
\(517\) −2625.00 −0.223302
\(518\) 0 0
\(519\) 19747.0 1.67013
\(520\) 0 0
\(521\) 9040.50 15658.6i 0.760214 1.31673i −0.182526 0.983201i \(-0.558427\pi\)
0.942740 0.333528i \(-0.108239\pi\)
\(522\) 0 0
\(523\) 10188.5 + 17647.0i 0.851839 + 1.47543i 0.879546 + 0.475813i \(0.157846\pi\)
−0.0277071 + 0.999616i \(0.508821\pi\)
\(524\) 0 0
\(525\) −1862.00 + 9675.24i −0.154789 + 0.804308i
\(526\) 0 0
\(527\) −1543.50 2673.42i −0.127582 0.220979i
\(528\) 0 0
\(529\) −6557.00 + 11357.1i −0.538917 + 0.933431i
\(530\) 0 0
\(531\) 2310.00 0.188786
\(532\) 0 0
\(533\) −4900.00 −0.398204
\(534\) 0 0
\(535\) 1599.50 2770.42i 0.129257 0.223879i
\(536\) 0 0
\(537\) 11385.5 + 19720.3i 0.914936 + 1.58472i
\(538\) 0 0
\(539\) −1592.50 636.529i −0.127261 0.0508668i
\(540\) 0 0
\(541\) 3096.50 + 5363.30i 0.246079 + 0.426222i 0.962435 0.271514i \(-0.0875243\pi\)
−0.716355 + 0.697736i \(0.754191\pi\)
\(542\) 0 0
\(543\) 5537.00 9590.37i 0.437597 0.757941i
\(544\) 0 0
\(545\) −7875.00 −0.618950
\(546\) 0 0
\(547\) 18464.0 1.44326 0.721630 0.692279i \(-0.243393\pi\)
0.721630 + 0.692279i \(0.243393\pi\)
\(548\) 0 0
\(549\) 4543.00 7868.71i 0.353170 0.611709i
\(550\) 0 0
\(551\) 1421.00 + 2461.24i 0.109867 + 0.190295i
\(552\) 0 0
\(553\) −360.500 + 1873.21i −0.0277216 + 0.144045i
\(554\) 0 0
\(555\) 5365.50 + 9293.32i 0.410365 + 0.710774i
\(556\) 0 0
\(557\) 4706.50 8151.90i 0.358027 0.620120i −0.629604 0.776916i \(-0.716783\pi\)
0.987631 + 0.156796i \(0.0501164\pi\)
\(558\) 0 0
\(559\) −1736.00 −0.131351
\(560\) 0 0
\(561\) 735.000 0.0553150
\(562\) 0 0
\(563\) 1599.50 2770.42i 0.119735 0.207387i −0.799928 0.600097i \(-0.795129\pi\)
0.919663 + 0.392709i \(0.128462\pi\)
\(564\) 0 0
\(565\) 5383.00 + 9323.63i 0.400822 + 0.694244i
\(566\) 0 0
\(567\) −14682.5 + 5086.17i −1.08749 + 0.376718i
\(568\) 0 0
\(569\) −10791.5 18691.4i −0.795085 1.37713i −0.922785 0.385314i \(-0.874093\pi\)
0.127701 0.991813i \(-0.459240\pi\)
\(570\) 0 0
\(571\) 10133.5 17551.7i 0.742686 1.28637i −0.208582 0.978005i \(-0.566885\pi\)
0.951268 0.308365i \(-0.0997819\pi\)
\(572\) 0 0
\(573\) 17899.0 1.30496
\(574\) 0 0
\(575\) −12084.0 −0.876413
\(576\) 0 0
\(577\) −6975.50 + 12081.9i −0.503282 + 0.871710i 0.496711 + 0.867916i \(0.334541\pi\)
−0.999993 + 0.00379418i \(0.998792\pi\)
\(578\) 0 0
\(579\) −1389.50 2406.68i −0.0997334 0.172743i
\(580\) 0 0
\(581\) 15288.0 + 13239.8i 1.09166 + 0.945403i
\(582\) 0 0
\(583\) −757.500 1312.03i −0.0538121 0.0932053i
\(584\) 0 0
\(585\) 1078.00 1867.15i 0.0761877 0.131961i
\(586\) 0 0
\(587\) 20972.0 1.47463 0.737314 0.675550i \(-0.236094\pi\)
0.737314 + 0.675550i \(0.236094\pi\)
\(588\) 0 0
\(589\) 7203.00 0.503895
\(590\) 0 0
\(591\) 10199.0 17665.2i 0.709866 1.22952i
\(592\) 0 0
\(593\) 94.5000 + 163.679i 0.00654410 + 0.0113347i 0.869279 0.494322i \(-0.164584\pi\)
−0.862735 + 0.505657i \(0.831250\pi\)
\(594\) 0 0
\(595\) 2058.00 + 1782.28i 0.141798 + 0.122801i
\(596\) 0 0
\(597\) −11686.5 20241.6i −0.801167 1.38766i
\(598\) 0 0
\(599\) −5140.50 + 8903.61i −0.350643 + 0.607331i −0.986362 0.164589i \(-0.947370\pi\)
0.635719 + 0.771920i \(0.280704\pi\)
\(600\) 0 0
\(601\) −6090.00 −0.413338 −0.206669 0.978411i \(-0.566262\pi\)
−0.206669 + 0.978411i \(0.566262\pi\)
\(602\) 0 0
\(603\) −9130.00 −0.616588
\(604\) 0 0
\(605\) 4571.00 7917.20i 0.307170 0.532033i
\(606\) 0 0
\(607\) 2474.50 + 4285.96i 0.165464 + 0.286593i 0.936820 0.349812i \(-0.113754\pi\)
−0.771356 + 0.636404i \(0.780421\pi\)
\(608\) 0 0
\(609\) −7105.00 + 2461.24i −0.472757 + 0.163768i
\(610\) 0 0
\(611\) −3675.00 6365.29i −0.243330 0.421460i
\(612\) 0 0
\(613\) 7898.50 13680.6i 0.520420 0.901394i −0.479298 0.877652i \(-0.659109\pi\)
0.999718 0.0237416i \(-0.00755791\pi\)
\(614\) 0 0
\(615\) −17150.0 −1.12448
\(616\) 0 0
\(617\) −9378.00 −0.611903 −0.305951 0.952047i \(-0.598975\pi\)
−0.305951 + 0.952047i \(0.598975\pi\)
\(618\) 0 0
\(619\) −12176.5 + 21090.3i −0.790654 + 1.36945i 0.134908 + 0.990858i \(0.456926\pi\)
−0.925562 + 0.378595i \(0.876407\pi\)
\(620\) 0 0
\(621\) −2782.50 4819.43i −0.179803 0.311429i
\(622\) 0 0
\(623\) 1151.50 5983.37i 0.0740512 0.384781i
\(624\) 0 0
\(625\) 174.500 + 302.243i 0.0111680 + 0.0193435i
\(626\) 0 0
\(627\) −857.500 + 1485.23i −0.0546176 + 0.0946005i
\(628\) 0 0
\(629\) −4599.00 −0.291533
\(630\) 0 0
\(631\) 12640.0 0.797449 0.398725 0.917071i \(-0.369453\pi\)
0.398725 + 0.917071i \(0.369453\pi\)
\(632\) 0 0
\(633\) −6230.00 + 10790.7i −0.391185 + 0.677553i
\(634\) 0 0
\(635\) 252.000 + 436.477i 0.0157485 + 0.0272772i
\(636\) 0 0
\(637\) −686.000 4752.75i −0.0426692 0.295621i
\(638\) 0 0
\(639\) −4752.00 8230.71i −0.294188 0.509549i
\(640\) 0 0
\(641\) 520.500 901.532i 0.0320726 0.0555513i −0.849544 0.527518i \(-0.823123\pi\)
0.881616 + 0.471967i \(0.156456\pi\)
\(642\) 0 0
\(643\) −9548.00 −0.585593 −0.292797 0.956175i \(-0.594586\pi\)
−0.292797 + 0.956175i \(0.594586\pi\)
\(644\) 0 0
\(645\) −6076.00 −0.370918
\(646\) 0 0
\(647\) −1620.50 + 2806.79i −0.0984674 + 0.170551i −0.911050 0.412295i \(-0.864727\pi\)
0.812583 + 0.582845i \(0.198061\pi\)
\(648\) 0 0
\(649\) −262.500 454.663i −0.0158768 0.0274994i
\(650\) 0 0
\(651\) −3601.50 + 18713.9i −0.216826 + 1.12666i
\(652\) 0 0
\(653\) 4426.50 + 7666.92i 0.265272 + 0.459464i 0.967635 0.252355i \(-0.0812051\pi\)
−0.702363 + 0.711819i \(0.747872\pi\)
\(654\) 0 0
\(655\) 7521.50 13027.6i 0.448686 0.777147i
\(656\) 0 0
\(657\) −24486.0 −1.45402
\(658\) 0 0
\(659\) −7044.00 −0.416381 −0.208191 0.978088i \(-0.566757\pi\)
−0.208191 + 0.978088i \(0.566757\pi\)
\(660\) 0 0
\(661\) 6044.50 10469.4i 0.355679 0.616054i −0.631555 0.775331i \(-0.717583\pi\)
0.987234 + 0.159277i \(0.0509163\pi\)
\(662\) 0 0
\(663\) 1029.00 + 1782.28i 0.0602761 + 0.104401i
\(664\) 0 0
\(665\) −6002.50 + 2079.33i −0.350026 + 0.121252i
\(666\) 0 0
\(667\) −4611.00 7986.49i −0.267674 0.463625i
\(668\) 0 0
\(669\) 4900.00 8487.05i 0.283176 0.490476i
\(670\) 0 0
\(671\) −2065.00 −0.118805
\(672\) 0 0
\(673\) 982.000 0.0562456 0.0281228 0.999604i \(-0.491047\pi\)
0.0281228 + 0.999604i \(0.491047\pi\)
\(674\) 0 0
\(675\) 1330.00 2303.63i 0.0758396 0.131358i
\(676\) 0 0
\(677\) 15256.5 + 26425.0i 0.866108 + 1.50014i 0.865943 + 0.500143i \(0.166719\pi\)
0.000164659 1.00000i \(0.499948\pi\)
\(678\) 0 0
\(679\) 12348.0 + 10693.7i 0.697898 + 0.604397i
\(680\) 0 0
\(681\) 7717.50 + 13367.1i 0.434266 + 0.752171i
\(682\) 0 0
\(683\) 5737.50 9937.64i 0.321434 0.556740i −0.659350 0.751836i \(-0.729169\pi\)
0.980784 + 0.195096i \(0.0625019\pi\)
\(684\) 0 0
\(685\) −7875.00 −0.439253
\(686\) 0 0
\(687\) −2009.00 −0.111569
\(688\) 0 0
\(689\) 2121.00 3673.68i 0.117277 0.203129i
\(690\) 0 0
\(691\) −14157.5 24521.5i −0.779416 1.34999i −0.932279 0.361741i \(-0.882182\pi\)
0.152862 0.988248i \(-0.451151\pi\)
\(692\) 0 0
\(693\) −1540.00 1333.68i −0.0844152 0.0731057i
\(694\) 0 0
\(695\) 882.000 + 1527.67i 0.0481384 + 0.0833781i
\(696\) 0 0
\(697\) 3675.00 6365.29i 0.199714 0.345915i
\(698\) 0 0
\(699\) −32109.0 −1.73744
\(700\) 0 0
\(701\) 10614.0 0.571876 0.285938 0.958248i \(-0.407695\pi\)
0.285938 + 0.958248i \(0.407695\pi\)
\(702\) 0 0
\(703\) 5365.50 9293.32i 0.287857 0.498583i
\(704\) 0 0
\(705\) −12862.5 22278.5i −0.687134 1.19015i
\(706\) 0 0
\(707\) 24132.5 8359.74i 1.28373 0.444697i
\(708\) 0 0
\(709\) −5149.50 8919.20i −0.272769 0.472451i 0.696801 0.717265i \(-0.254606\pi\)
−0.969570 + 0.244814i \(0.921273\pi\)
\(710\) 0 0
\(711\) −1133.00 + 1962.41i −0.0597621 + 0.103511i
\(712\) 0 0
\(713\) −23373.0 −1.22767
\(714\) 0 0
\(715\) −490.000 −0.0256293
\(716\) 0 0
\(717\) −5838.00 + 10111.7i −0.304078 + 0.526679i
\(718\) 0 0
\(719\) 16264.5 + 28170.9i 0.843621 + 1.46119i 0.886813 + 0.462128i \(0.152914\pi\)
−0.0431924 + 0.999067i \(0.513753\pi\)
\(720\) 0 0
\(721\) −2376.50 + 12348.7i −0.122754 + 0.637847i
\(722\) 0 0
\(723\) −11931.5 20666.0i −0.613744 1.06304i
\(724\) 0 0
\(725\) 2204.00 3817.44i 0.112903 0.195553i
\(726\) 0 0
\(727\) −29456.0 −1.50270 −0.751350 0.659904i \(-0.770597\pi\)
−0.751350 + 0.659904i \(0.770597\pi\)
\(728\) 0 0
\(729\) −11843.0 −0.601687
\(730\) 0 0
\(731\) 1302.00 2255.13i 0.0658772 0.114103i
\(732\) 0 0
\(733\) −13933.5 24133.5i −0.702109 1.21609i −0.967725 0.252009i \(-0.918909\pi\)
0.265616 0.964079i \(-0.414425\pi\)
\(734\) 0 0
\(735\) −2401.00 16634.6i −0.120493 0.834799i
\(736\) 0 0
\(737\) 1037.50 + 1797.00i 0.0518546 + 0.0898147i
\(738\) 0 0
\(739\) 9769.50 16921.3i 0.486302 0.842299i −0.513574 0.858045i \(-0.671679\pi\)
0.999876 + 0.0157460i \(0.00501231\pi\)
\(740\) 0 0
\(741\) −4802.00 −0.238065
\(742\) 0 0
\(743\) −1248.00 −0.0616214 −0.0308107 0.999525i \(-0.509809\pi\)
−0.0308107 + 0.999525i \(0.509809\pi\)
\(744\) 0 0
\(745\) 703.500 1218.50i 0.0345963 0.0599226i
\(746\) 0 0
\(747\) 12012.0 + 20805.4i 0.588348 + 1.01905i
\(748\) 0 0
\(749\) 1599.50 8311.25i 0.0780300 0.405456i
\(750\) 0 0
\(751\) 14046.5 + 24329.3i 0.682509 + 1.18214i 0.974213 + 0.225631i \(0.0724444\pi\)
−0.291704 + 0.956509i \(0.594222\pi\)
\(752\) 0 0
\(753\) 16660.0 28856.0i 0.806274 1.39651i
\(754\) 0 0
\(755\) −11333.0 −0.546292
\(756\) 0 0
\(757\) 35954.0 1.72625 0.863124 0.504991i \(-0.168504\pi\)
0.863124 + 0.504991i \(0.168504\pi\)
\(758\) 0 0
\(759\) 2782.50 4819.43i 0.133068 0.230480i
\(760\) 0 0
\(761\) 430.500 + 745.648i 0.0205067 + 0.0355187i 0.876097 0.482136i \(-0.160139\pi\)
−0.855590 + 0.517654i \(0.826805\pi\)
\(762\) 0 0
\(763\) −19687.5 + 6819.95i −0.934122 + 0.323589i
\(764\) 0 0
\(765\) 1617.00 + 2800.73i 0.0764219 + 0.132367i
\(766\) 0 0
\(767\) 735.000 1273.06i 0.0346014 0.0599315i
\(768\) 0 0
\(769\) 24710.0 1.15873 0.579366 0.815067i \(-0.303300\pi\)
0.579366 + 0.815067i \(0.303300\pi\)
\(770\) 0 0
\(771\) 5635.00 0.263216
\(772\) 0 0
\(773\) −8249.50 + 14288.6i −0.383847 + 0.664843i −0.991609 0.129277i \(-0.958734\pi\)
0.607761 + 0.794120i \(0.292068\pi\)
\(774\) 0 0
\(775\) −5586.00 9675.24i −0.258910 0.448445i
\(776\) 0 0
\(777\) 21462.0 + 18586.6i 0.990920 + 0.858162i
\(778\) 0 0
\(779\) 8575.00 + 14852.3i 0.394392 + 0.683107i
\(780\) 0 0
\(781\) −1080.00 + 1870.61i −0.0494820 + 0.0857053i
\(782\) 0 0
\(783\) 2030.00 0.0926517
\(784\) 0 0
\(785\) 4753.00 0.216104
\(786\) 0 0
\(787\) 8235.50 14264.3i 0.373016 0.646083i −0.617012 0.786954i \(-0.711657\pi\)
0.990028 + 0.140871i \(0.0449902\pi\)
\(788\) 0 0
\(789\) 899.500 + 1557.98i 0.0405869 + 0.0702985i
\(790\) 0 0
\(791\) 21532.0 + 18647.3i 0.967876 + 0.838205i
\(792\) 0 0
\(793\) −2891.00 5007.36i −0.129461 0.224233i
\(794\) 0 0
\(795\) 7423.50 12857.9i 0.331175 0.573613i
\(796\) 0 0
\(797\) −36470.0 −1.62087 −0.810435 0.585828i \(-0.800769\pi\)
−0.810435 + 0.585828i \(0.800769\pi\)
\(798\) 0 0
\(799\) 11025.0 0.488156
\(800\) 0 0
\(801\) 3619.00 6268.29i 0.159639 0.276503i
\(802\) 0 0
\(803\) 2782.50 + 4819.43i 0.122282 + 0.211798i
\(804\) 0 0
\(805\) 19477.5 6747.20i 0.852785 0.295413i
\(806\) 0 0
\(807\) 12568.5 + 21769.3i 0.548243 + 0.949585i
\(808\) 0 0
\(809\) −17875.5 + 30961.3i −0.776847 + 1.34554i 0.156904 + 0.987614i \(0.449849\pi\)
−0.933751 + 0.357924i \(0.883485\pi\)
\(810\) 0 0
\(811\) 16492.0 0.714072 0.357036 0.934091i \(-0.383787\pi\)
0.357036 + 0.934091i \(0.383787\pi\)
\(812\) 0 0
\(813\) 9751.00 0.420643
\(814\) 0 0
\(815\) −1634.50 + 2831.04i −0.0702504 + 0.121677i
\(816\) 0 0
\(817\) 3038.00 + 5261.97i 0.130093 + 0.225328i
\(818\) 0 0
\(819\) 1078.00 5601.45i 0.0459931 0.238987i
\(820\) 0 0
\(821\) 20736.5 + 35916.7i 0.881497 + 1.52680i 0.849677 + 0.527304i \(0.176797\pi\)
0.0318198 + 0.999494i \(0.489870\pi\)
\(822\) 0 0
\(823\) −12532.5 + 21706.9i −0.530809 + 0.919387i 0.468545 + 0.883440i \(0.344778\pi\)
−0.999354 + 0.0359479i \(0.988555\pi\)
\(824\) 0 0
\(825\) 2660.00 0.112254
\(826\) 0 0
\(827\) −9732.00 −0.409208 −0.204604 0.978845i \(-0.565591\pi\)
−0.204604 + 0.978845i \(0.565591\pi\)
\(828\) 0 0
\(829\) −13877.5 + 24036.5i −0.581406 + 1.00702i 0.413907 + 0.910319i \(0.364164\pi\)
−0.995313 + 0.0967055i \(0.969170\pi\)
\(830\) 0 0
\(831\) 1452.50 + 2515.80i 0.0606338 + 0.105021i
\(832\) 0 0
\(833\) 6688.50 + 2673.42i 0.278203 + 0.111199i
\(834\) 0 0
\(835\) 4214.00 + 7298.86i 0.174648 + 0.302500i
\(836\) 0 0
\(837\) 2572.50 4455.70i 0.106235 0.184004i
\(838\) 0 0
\(839\) −21112.0 −0.868733 −0.434367 0.900736i \(-0.643028\pi\)
−0.434367 + 0.900736i \(0.643028\pi\)
\(840\) 0 0
\(841\) −21025.0 −0.862069
\(842\) 0 0
\(843\) −17339.0 + 30032.0i −0.708407 + 1.22700i
\(844\) 0 0
\(845\) 7003.50 + 12130.4i 0.285122 + 0.493845i
\(846\) 0 0
\(847\) 4571.00 23751.6i 0.185433 0.963536i
\(848\) 0 0
\(849\) 14969.5 + 25927.9i 0.605126 + 1.04811i
\(850\) 0 0
\(851\) −17410.5 + 30155.9i −0.701321 + 1.21472i
\(852\) 0 0
\(853\) −21238.0 −0.852492 −0.426246 0.904607i \(-0.640164\pi\)
−0.426246 + 0.904607i \(0.640164\pi\)
\(854\) 0 0
\(855\) −7546.00 −0.301834
\(856\) 0 0
\(857\) 17804.5 30838.3i 0.709673 1.22919i −0.255305 0.966861i \(-0.582176\pi\)
0.964978 0.262330i \(-0.0844908\pi\)
\(858\) 0 0
\(859\) 1088.50 + 1885.34i 0.0432353 + 0.0748858i 0.886833 0.462090i \(-0.152900\pi\)
−0.843598 + 0.536975i \(0.819567\pi\)
\(860\) 0 0
\(861\) −42875.0 + 14852.3i −1.69707 + 0.587882i
\(862\) 0 0
\(863\) −16123.5 27926.7i −0.635980 1.10155i −0.986307 0.164921i \(-0.947263\pi\)
0.350327 0.936627i \(-0.386070\pi\)
\(864\) 0 0
\(865\) 9873.50 17101.4i 0.388103 0.672214i
\(866\) 0 0
\(867\) 31304.0 1.22623
\(868\) 0 0
\(869\) 515.000 0.0201038
\(870\) 0 0
\(871\) −2905.00 + 5031.61i −0.113011 + 0.195740i
\(872\) 0 0
\(873\) 9702.00 + 16804.4i 0.376132 + 0.651479i
\(874\) 0 0
\(875\) 19698.0 + 17059.0i 0.761045 + 0.659084i
\(876\) 0 0
\(877\) −13815.5 23929.1i −0.531946 0.921357i −0.999305 0.0372891i \(-0.988128\pi\)
0.467359 0.884068i \(-0.345206\pi\)
\(878\) 0 0
\(879\) 27097.0 46933.4i 1.03977 1.80094i
\(880\) 0 0
\(881\) 24402.0 0.933172 0.466586 0.884476i \(-0.345484\pi\)
0.466586 + 0.884476i \(0.345484\pi\)
\(882\) 0 0
\(883\) 19612.0 0.747448 0.373724 0.927540i \(-0.378081\pi\)
0.373724 + 0.927540i \(0.378081\pi\)
\(884\) 0 0
\(885\) 2572.50 4455.70i 0.0977103 0.169239i
\(886\) 0 0
\(887\) 1130.50 + 1958.08i 0.0427942 + 0.0741218i 0.886629 0.462481i \(-0.153041\pi\)
−0.843835 + 0.536603i \(0.819707\pi\)
\(888\) 0 0
\(889\) 1008.00 + 872.954i 0.0380284 + 0.0329335i
\(890\) 0 0
\(891\) 2097.50 + 3632.98i 0.0788652 + 0.136599i
\(892\) 0 0
\(893\) −12862.5 + 22278.5i −0.482001 + 0.834851i
\(894\) 0 0
\(895\) 22771.0 0.850448
\(896\) 0 0
\(897\) 15582.0 0.580009
\(898\) 0 0
\(899\) 4263.00 7383.73i 0.158152 0.273928i
\(900\) 0 0
\(901\) 3181.50 + 5510.52i 0.117637 + 0.203754i
\(902\) 0 0
\(903\) −15190.0 + 5261.97i −0.559791 + 0.193917i
\(904\) 0 0
\(905\) −5537.00 9590.37i −0.203377 0.352259i
\(906\) 0 0
\(907\) −11916.5 + 20640.0i −0.436252 + 0.755611i −0.997397 0.0721066i \(-0.977028\pi\)
0.561145 + 0.827718i \(0.310361\pi\)
\(908\) 0 0
\(909\) 30338.0 1.10698
\(910\) 0 0
\(911\) −31824.0 −1.15738 −0.578692 0.815546i \(-0.696437\pi\)
−0.578692 + 0.815546i \(0.696437\pi\)
\(912\) 0 0
\(913\) 2730.00 4728.50i 0.0989593 0.171402i
\(914\) 0 0
\(915\) −10118.5 17525.8i −0.365582 0.633206i
\(916\) 0 0
\(917\) 7521.50 39082.9i 0.270863 1.40745i
\(918\) 0 0
\(919\) −8409.50 14565.7i −0.301854 0.522826i 0.674702 0.738090i \(-0.264272\pi\)
−0.976556 + 0.215264i \(0.930939\pi\)
\(920\) 0 0
\(921\) 25774.0 44641.9i 0.922130 1.59718i
\(922\) 0 0
\(923\) −6048.00 −0.215680
\(924\) 0 0
\(925\) −16644.0 −0.591623
\(926\) 0 0
\(927\) −7469.00 + 12936.7i −0.264632 + 0.458357i
\(928\) 0 0
\(929\) −899.500 1557.98i −0.0317671 0.0550222i 0.849705 0.527259i \(-0.176780\pi\)
−0.881472 + 0.472237i \(0.843447\pi\)
\(930\) 0 0
\(931\) −13205.5 + 10396.6i −0.464869 + 0.365989i
\(932\) 0 0
\(933\) −34912.5 60470.2i −1.22506 2.12187i
\(934\) 0 0
\(935\) 367.500 636.529i 0.0128540 0.0222639i
\(936\) 0 0
\(937\) 14154.0 0.493480 0.246740 0.969082i \(-0.420641\pi\)
0.246740 + 0.969082i \(0.420641\pi\)
\(938\) 0 0
\(939\) 33271.0 1.15629
\(940\) 0 0
\(941\) −6023.50 + 10433.0i −0.208672 + 0.361431i −0.951296 0.308277i \(-0.900247\pi\)
0.742624 + 0.669708i \(0.233581\pi\)
\(942\) 0 0
\(943\) −27825.0 48194.3i −0.960877 1.66429i
\(944\) 0 0
\(945\) −857.500 + 4455.70i −0.0295180 + 0.153380i
\(946\) 0 0
\(947\) −12189.5 21112.8i −0.418274 0.724472i 0.577492 0.816396i \(-0.304031\pi\)
−0.995766 + 0.0919245i \(0.970698\pi\)
\(948\) 0 0
\(949\) −7791.00 + 13494.4i −0.266498 + 0.461588i
\(950\) 0 0
\(951\) 24339.0 0.829912
\(952\) 0 0
\(953\) −52330.0 −1.77874 −0.889368 0.457192i \(-0.848855\pi\)
−0.889368 + 0.457192i \(0.848855\pi\)
\(954\) 0 0
\(955\) 8949.50 15501.0i 0.303245 0.525236i
\(956\) 0 0
\(957\) 1015.00 + 1758.03i 0.0342845 + 0.0593825i
\(958\) 0 0
\(959\) −19687.5 + 6819.95i −0.662922 + 0.229643i
\(960\) 0 0
\(961\) 4091.00 + 7085.82i 0.137323 + 0.237851i
\(962\) 0 0
\(963\) 5027.00 8707.02i 0.168217 0.291360i
\(964\) 0 0
\(965\) −2779.00 −0.0927038
\(966\) 0 0
\(967\) 12416.0 0.412897 0.206449 0.978457i \(-0.433809\pi\)
0.206449 + 0.978457i \(0.433809\pi\)
\(968\) 0 0
\(969\) 3601.50 6237.98i 0.119398 0.206804i
\(970\) 0 0
\(971\) 18406.5 + 31881.0i 0.608334 + 1.05367i 0.991515 + 0.129993i \(0.0414954\pi\)
−0.383181 + 0.923673i \(0.625171\pi\)
\(972\) 0 0
\(973\) 3528.00 + 3055.34i 0.116241 + 0.100668i
\(974\) 0 0
\(975\) 3724.00 + 6450.16i 0.122321 + 0.211867i
\(976\) 0 0
\(977\) −17497.5 + 30306.6i −0.572973 + 0.992418i 0.423286 + 0.905996i \(0.360877\pi\)
−0.996259 + 0.0864221i \(0.972457\pi\)
\(978\) 0 0
\(979\) −1645.00 −0.0537022
\(980\) 0 0
\(981\) −24750.0 −0.805511
\(982\) 0 0
\(983\) −7150.50 + 12385.0i −0.232010 + 0.401853i −0.958399 0.285430i \(-0.907863\pi\)
0.726390 + 0.687283i \(0.241197\pi\)
\(984\) 0 0
\(985\) −10199.0 17665.2i −0.329916 0.571431i
\(986\) 0 0
\(987\) −51450.0 44557.0i −1.65924 1.43695i
\(988\) 0 0
\(989\) −9858.00 17074.6i −0.316953 0.548978i
\(990\) 0 0
\(991\) −1332.50 + 2307.96i −0.0427127 + 0.0739805i −0.886591 0.462553i \(-0.846933\pi\)
0.843879 + 0.536534i \(0.180267\pi\)
\(992\) 0 0
\(993\) 23387.0 0.747396
\(994\) 0 0
\(995\) −23373.0 −0.744697
\(996\) 0 0
\(997\) −12435.5 + 21538.9i −0.395021 + 0.684197i −0.993104 0.117237i \(-0.962596\pi\)
0.598083 + 0.801434i \(0.295929\pi\)
\(998\) 0 0
\(999\) −3832.50 6638.08i −0.121376 0.210230i
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 112.4.i.c.81.1 2
4.3 odd 2 7.4.c.a.4.1 yes 2
7.2 even 3 inner 112.4.i.c.65.1 2
7.3 odd 6 784.4.a.r.1.1 1
7.4 even 3 784.4.a.b.1.1 1
8.3 odd 2 448.4.i.f.193.1 2
8.5 even 2 448.4.i.a.193.1 2
12.11 even 2 63.4.e.b.46.1 2
20.3 even 4 175.4.k.a.74.1 4
20.7 even 4 175.4.k.a.74.2 4
20.19 odd 2 175.4.e.a.151.1 2
28.3 even 6 49.4.a.c.1.1 1
28.11 odd 6 49.4.a.d.1.1 1
28.19 even 6 49.4.c.a.30.1 2
28.23 odd 6 7.4.c.a.2.1 2
28.27 even 2 49.4.c.a.18.1 2
56.37 even 6 448.4.i.a.65.1 2
56.51 odd 6 448.4.i.f.65.1 2
84.11 even 6 441.4.a.d.1.1 1
84.23 even 6 63.4.e.b.37.1 2
84.47 odd 6 441.4.e.k.226.1 2
84.59 odd 6 441.4.a.e.1.1 1
84.83 odd 2 441.4.e.k.361.1 2
140.23 even 12 175.4.k.a.149.2 4
140.39 odd 6 1225.4.a.c.1.1 1
140.59 even 6 1225.4.a.d.1.1 1
140.79 odd 6 175.4.e.a.51.1 2
140.107 even 12 175.4.k.a.149.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
7.4.c.a.2.1 2 28.23 odd 6
7.4.c.a.4.1 yes 2 4.3 odd 2
49.4.a.c.1.1 1 28.3 even 6
49.4.a.d.1.1 1 28.11 odd 6
49.4.c.a.18.1 2 28.27 even 2
49.4.c.a.30.1 2 28.19 even 6
63.4.e.b.37.1 2 84.23 even 6
63.4.e.b.46.1 2 12.11 even 2
112.4.i.c.65.1 2 7.2 even 3 inner
112.4.i.c.81.1 2 1.1 even 1 trivial
175.4.e.a.51.1 2 140.79 odd 6
175.4.e.a.151.1 2 20.19 odd 2
175.4.k.a.74.1 4 20.3 even 4
175.4.k.a.74.2 4 20.7 even 4
175.4.k.a.149.1 4 140.107 even 12
175.4.k.a.149.2 4 140.23 even 12
441.4.a.d.1.1 1 84.11 even 6
441.4.a.e.1.1 1 84.59 odd 6
441.4.e.k.226.1 2 84.47 odd 6
441.4.e.k.361.1 2 84.83 odd 2
448.4.i.a.65.1 2 56.37 even 6
448.4.i.a.193.1 2 8.5 even 2
448.4.i.f.65.1 2 56.51 odd 6
448.4.i.f.193.1 2 8.3 odd 2
784.4.a.b.1.1 1 7.4 even 3
784.4.a.r.1.1 1 7.3 odd 6
1225.4.a.c.1.1 1 140.39 odd 6
1225.4.a.d.1.1 1 140.59 even 6